In this paper, we introduce a new class of parameterized problems, which we call XALP: the class of all parameterized problems that can be solved in $f(k)n^{O(1)}$ time and $f(k)\log n$ space on a non-deterministic Turing Machine with access to an auxiliary stack (with only top element lookup allowed). Various natural problems on `tree-structured graphs' are complete for this class: we show that List Colouring and All-or-Nothing Flow parameterized by treewidth are XALP-complete. Moreover, Independent Set and Dominating Set parameterized by treewidth divided by $\log n$, and Max Cut parameterized by cliquewidth are also XALP-complete. Besides finding a `natural home' for these problems, we also pave the road for future reductions. We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most $f(k)n^{O(1)}$ and use $f(k)\log n$ space. Moreover, we introduce `tree-shaped' variants of Weighted CNF-Satisfiability and Multicolour Clique that are XALP-complete.

翻译：本文引入了一个新的参数化问题类，我们称之为XALP：该类包含所有可通过非确定性图灵机（允许访问辅助栈，仅支持栈顶元素查询）在$f(k)n^{O(1)}$时间和$f(k)\log n$空间内求解的参数化问题。若干关于"树结构图"的自然问题在此类中完备：我们证明了以树宽为参数的列表染色问题和全有或全无流问题属于XALP完备类。此外，以树宽除以$\log n$为参数的独立集问题和支配集问题，以及以团宽为参数的最大割问题同样属于XALP完备类。除了为这些问题提供"自然归属"外，我们还为后续归约研究铺平了道路。我们给出了XALP类的若干等价刻画，例如：XALP等价于可通过交替式图灵机求解的问题类，其运行过程的树规模至多为$f(k)n^{O(1)}$且空间使用量为$f(k)\log n$。此外，我们还引入了加权CNF可满足性和多色团问题的"树形"变种，这些变种属于XALP完备类。